Thermal, Structural, and Inflation Modeling of an Isotensoid ...

q c
, W/cm 2
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25 30
Time, s
The material “stack” is discretized into a surface node,
several interior nodes, and an inner node. Nodes are
uniformly spaced according to the total thickness of the
material stack and the chosen number of nodes, i max = 50. In
order to maintain uniform node spacing, the surface and
inner nodes are assigned thickness Δx/2 and the interior
nodes are assigned thickness Δx (see Figure 12). Nodal
energy balance equations are derived and explicitly solved
for nodal temperatures using a time-marching numerical
scheme. The following discussions derive expressions for
the nodal temperatures through the thickness as a function
of time.
Surface Node
The energy balance at the surface node is given by the
model below and Equation (15):
Figure 11 – Heating boundary condition.
5. THERMAL MODEL
Selecting suitable SIAD materials requires understanding of
the thermal environment throughout the envelope. The
transient temperature response within the multi-material
SIAD envelope, T(x,t), is computed with one-dimensional
heat conduction relations [13]. Three material layers are
considered: an outer layer of elastomeric coating, the
structural fabric, and an inner layer of elastomeric coating.
Figure 12 shows the three material layers and the
discretization scheme.
!x/2!
Surface Coating
Structural Fabric
Flow Direction
!x!
Surface Node (i = 1)
Interior Nodes (2 ! i ! i max -1)
Inner Coating
!x/2!
Inner Node (i = i max )
IAD Interior
Figure 12 – Notional discrete model of three-layer IAD
envelope.
where q ˙ c is the convective heating from the boundary layer,
ε is the coating emissivity, σ is the Stephan-Boltzmann
! constant, ρ is the fabric density, and c p is the fabric specific
!
heat Convective at node 1. The Radiation term ρc p Heat is often conducted referred to as the
material heating from thermal -
Heat stored
into mass. space The - away temperature from outer notation =
in outer is slab as
boundary layer from surface slab
follows: refers to the temperature at node 1 to the
!
8
Convective
heating from
boundary layer
Energy added
to slab i over
time !t
q ˙ c "#$ T 4n 4n
( 1 "T % ) " C 1 T n n
( 1 "T 2 ) =
T 1 4n
&c p 'x
n
T +1 n
Energy
( 1 "T 1 )
transferred 2't -
T 1
n +1
(15)
fourth power at the n th timestep, refers to the
temperature at node 1 at the n th + 1 timestep, and so forth.
Convective Radiation Heat conducted
The
!
conductance term, C 1 , is a function of the material
heating from -
Heat stored
into space - away from outer =
properties boundary layer and the from current surface and neighboring slab nodes: in outer slab
!
#
C 1 = %
$
Radiation
into space
from surface
"x /2
k 1
+
"x /2
k 2
&
(
'
)1
= 2 # 1
+ 1 &
% (
"x $ k 1 k 2 '
Equation (16) is applicable to multi-material layups because
it ensures energy conservation between adjacent nodes with
! different material properties. This can be the case at the
surface node when the thickness of the surface coating is
much smaller than the thickness of the structural fabric.
Equation (15) can be solved for the surface temperature at
timestep n + 1:
n
T +1 1 = T n 2"t
1 +
# 1 c p1 "x q ˙ c $% 1 & T 4n 4n
1 $T '
[
$ 2 ( 1
+ 1 +
* -
"x ) k 1 k 2 ,
Heat conducted
away from outer
slab
- - =
=
from slab i-1
)1
( )
$1
T n n
( 1 $T 2 ) .
0
/
0
Heat stored
in outer slab
Energy
transferred
to slab i-1
(16)
(17)